[Proof of Theorem  REF (Rigidity theorem of \(c_{B}\)  and \(c_{\beta }\) )] If \(v(\Omega ) = \infty \) , then the inequality is trivial. By [1]}, \(c_B(z_0) = 0\)  if and only if \(c_B \equiv 0\) , and thus if and only if \(\Omega = \mathbb {C}_{\infty } \setminus P\)  where \(P \in \mathcal {N}_B\)  by definition.
